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(countable_image
(countable_image 0
(countable_image-2 nil 3322415765
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(lemma "inj_inv_alt[(S!1), (image(f!1)(S!1))]"
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nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (expand* "restrict" "image" "image")
(("2" (skolem!) (("2" (inst?) nil nil)) nil)) nil))
nil))
nil)
("2" (expand* "restrict" "surjective?")
(("2" (skolem-typepred)
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(("2" (skolem!)
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nil))
nil)
("3" (expand* "image" "image")
(("3" (skolem!) (("3" (inst?) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
((is_countable const-decl "bool" countability nil)
(R formal-type-decl nil countable_image nil)
(image const-decl "set[R]" function_image nil)
(surjective? const-decl "bool" functions nil)
(restrict const-decl "R" restrict nil)
(surjective_inverse_exists formula-decl nil function_inverse_def
nil)
(O const-decl "T3" function_props nil)
(g!1 skolem-const-decl "[(image(f!1)(S!1)) -> (S!1)]"
countable_image nil)
(f!2 skolem-const-decl "(injective?[(S!1), nat])" countable_image
nil)
(S!1 skolem-const-decl "set[D]" countable_image nil)
(f!1 skolem-const-decl "[D -> R]" countable_image nil)
(inj_inv_alt formula-decl nil function_inverse_def nil)
(inverse? const-decl "bool" function_inverse_def nil)
(image const-decl "set[R]" function_image nil)
(injective? const-decl "bool" functions nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals nil)
(int nonempty-type-eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(rational nonempty-type-from-decl nil rationals nil)
(rational_pred const-decl "[real -> boolean]" rationals nil)
(real nonempty-type-from-decl nil reals nil)
(real_pred const-decl "[number_field -> boolean]" reals nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(number nonempty-type-decl nil numbers nil)
(set type-eq-decl nil sets nil)
(D formal-type-decl nil countable_image nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
nil)
(countable_image-1 nil 3313909571
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(("" (case "is_finite(S!1)")
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(("1" (expand "injective?")
(("1" (propax) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (rewrite "countable_card" -1)
(("2" (split)
(("1" (propax) nil nil)
("2" (expand "is_countably_infinite")
(("2" (skosimp)
(("2" (typepred "f!2")
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(lemma
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(("1"
(lemma
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"restrict[D, ((S!1)), R](f!1)"
"g"
"g!1"))
(("1"
(expand "bijective?")
(("1"
(flatten)
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(hide-all-but (-1 -4 1))
(("1"
(expand "o")
(("1"
(expand "injective?")
(("1"
(skosimp*)
(("1"
(inst
-3
"g!1(x1!1)"
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(inst - "x1!1" "x2!1")
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(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (propax) nil nil))
nil))
nil))
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(("2" (skosimp*)
(("2" (expand "restrict")
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(expand "image")
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(expand "image")
(("2" (inst + "x1!1") nil nil))
nil))
nil))
nil))
nil))
nil))
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nil)
("2" (hide-all-but 1)
(("2" (expand "restrict")
(("2" (expand "surjective?")
(("2" (skosimp*)
(("2" (typepred "y!1")
(("2"
(expand "image")
(("2"
(expand "image")
(("2"
(skosimp*)
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(inst + "x!1")
(("2" (assert) nil nil))
nil))
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nil))
nil))
nil))
nil))
nil))
nil)
("3" (expand "image")
(("3" (expand "image")
(("3" (skosimp) (("3" (inst + "x1!1") nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(is_countable const-decl "bool" countability nil)
(is_countably_infinite const-decl "bool" countability nil))
shostak)))
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